Rajesh M. Darji
Department of Mathematics
Sarvajanik College of Engineering and
Technology
Surat-395 001, Gujarat
India
Munir G. Timol
Department of Mathematics
Veer Narmad South Gujarat University
Surt-395007, Gujarat
India
Similarity treatment for MHD free
convective boundary layer flow of a
class of Non-Newtonian Fluids
Deductive group symmetry treatment is appied to derive the similarity
transformations for the free convective boundary layer flow of a class of
non-Newtonian fluids past over a two-dimensional surface and flowing
under the influence of transverse magnetic field. Numerical solutions are
obtained for particular Non-Newtonian fluid model namely Prandtl Eyring
fluid, in a graphical form . The important physical quantities like velocity
distribution, skin friction coefficent and temperature variations are
discussed.
Keywords: Non-Newtonian fluid, Deuctive group symmetry, Similarity
transformation, MHD, Prandtl Eyring fluid, Boundary layer theory.
1.
INTRODUCTION
The classical theory of fluid mechanics is based upon
the hypothesis of a linear relationship between two
tensor components, shearing stress and rate of strain as,
τ = −µ
∂u
∂y
(1)
The fluids with properties different from that
described by equation (1), called Non-Newtonian fluids.
Flow of Non-Newtonian fluids has attained a great
success in the theory of fluid mechanics due to its
applications in biological sciences and industry.
Problems involving Non-Newtonian fluid models have
been studied by the many researchers since last decades.
Several techniques are found to analyze and derive the
solutions of governing equations. Some of these are
cited in Refs. [1–9]. The similarity technique plays an
important role in problem analysis, especially in the
boundary layer flows. The similarity method involves
the determination of similarity variables which reduce
the system of governing partial differential equations in
to ordinary differential equations. Indeed similarity
solution is the only class of more accurate solution for
the governing differential equations. In the present work
we have applied similarity treatment for the particular
boundary layer problem.
In the literature several information are available on
similarity solutions for the natural convective heat
transfer of a Non-Newtonian fluids [See 10-12].
Researchers have presented works on flow in an
electrically conducting Non-Newtonian fluid over a
stretching sheet [see 13-19]. At this point it is worth to
note that most of the work has been done for NonNewtonian power-law fluids; this is because of its
mathematical simplicity. However there are empirical
Received: May 2015, Accepted: February 2016
Correspondence to: Dr. Rajesh M. Darji
Department of Mathematics, Sarvajanik College of
Engineering and Technology, Gujarat, India
E-mail: [email protected]
doi:10.5937/fmet1602197D
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
Non-Newtonian fluid models based on functional
relationship between shear stress and rate of the strain
are available [20]. In present work we concentrate our
discussion on the similarity solution of steady laminar
natural convection flows of generalized Non-Newtonian
fluid. Such a class of fluids are severely omitted in the
analysis due to mathematical complicity of its nonlinear stress-strain relationship. Further, from these
charts, we noticed that all the similarity solutions
presented there in are derived either by adopting or by
ad-hoc assumption on similarity variables. In the
context of this work it is necessary to develop systema–
tical group transformation for similarity solution.
Hence, present work focused on deductive group
symmetry analysis based on general group of trans–
formations. The analysis is applied to the particular
problem of boundary layer theory. We investigate the
MHD boundary layer flow of a class of Non-Newtonian
fluids characterized by the property that its stress tensor
component τij can be related to the strain rate component
eij by the arbitrary continuous function of the type
Ω (τ ij , eij ) = 0
(2)
The similarity equations obtained are more general
and systematic along with auxiliary conditions.
Recently this method has been successfully applied to
various non-linear problems by Abd-el-Malek et al [21],
Adnan et al [22], Darji and Timol [23, 24].
2.
GOVERNING EQUATIONS
Consider the steady laminar natural convection flow of a
non-Newtonian fluid over a vertical permeable surface, in
the presence of transverse magnetic field. Consider the
vertical upward along the surface as positive x-direction,
and the origin is fixed (Fig. 1). The transverse
electrically conducting variable magnetic field of the
strength B(x) is applied normal to the X -axis. It is
assumed that the magnetic Reynolds number Rem is very
small; i.e. Rem = µ0σ L << 1 , where µ0 is the magnetic
permeability, L is the reference length of the plate and σ
is the electric conductivity. We neglect the induced
FME Transactions (2016) 44, 197-203 197
magnetic field, which is small in comparison with the
applied magnetic field. Also for a class of NonNewtonian fluids, the stress-strain relation, under the
boundary layer assumption can be found in the form of
arbitrary function with only non-vanishing component
τ y x . Then the relation (2) can be given by [see 20]
Substitute the values in (3) to (7) along with nondimensional stream function ψ ∗ ( x∗ , y ∗ ) such that
∂ψ ∗
∂ψ ∗
∗
and
v
=
−
, and dropping the asterisks
∂y ∗
∂x∗
(for simplicity), we get
u∗ =

∂u 
Ω τ y x ,  = 0
∂y 

2
2
∂ψ ∂ ψ ∂ψ ∂ ψ
∂
∂ψ
−
= (τ y x ) + θ − bB 2 ( x )
2
∂y ∂y∂x ∂x ∂y
∂y
∂y
∂ψ ∂θ ∂ψ ∂θ
1 ∂ 2θ
−
=
∂y ∂x ∂x ∂y 3Pr ∂y 2
(8)
(9)
(10)
in which b = σ / ρU ∞ , together with boundary conditions:
Figure 1. Schematic diagram of MHD natural convection

∂u 
Ω τ y x ,  = 0
∂y 

(3)
Using boundary layer approximations, the governing
equations for a class of non-Newtonian fluids are given
by [See 28, 29]:
Continuity Equation:
∂u ∂v
+
=0
∂x ∂y
(4)
Momentum Equation:
σ B ( x)
∂u
∂v 1 ∂
+v =
τ y x ) + g β 'θ −
u
(
∂x
∂y ρ ∂y
ρ
2
u
(5)
u
∂θ
∂θ
∂ 2θ
+v
=α' 2
∂x
∂y
∂y
(6)
Together with boundary conditions:
∀x, u = v = 0, θ = θ w ( x ) at y = 0 

∀x , u = θ = 0
as y → ∞ 
(7)
where α ' is thermal diffusivity, β ' is the volumetric
thermal expansion coefficient.
Introducing the following dimensionless quantities,
Gr
x =
x,
L
∗
1/2
 Re

y =
⋅ Gr 
3


∗
y
,
L
1/2
u∗ =
1/ 2
 Re 
Re =
U∞ L
ν
,
3.


 (11)


SIMILARITY ANALYSIS
We now seek some sort of transformation, namely,
similarity transformation which transforms the partial
differential Eqs (8) to (10) into the ordinary differential
equations along with appropriate auxiliary conditions (11).
To search this transformation, the one-parameter general
deductive group of transformations is introduced as:
G : Q ( a ) = ℵQ ( a ) s + ℜQ
(12)
where Q stands for x, y,ψ ,θ ,τ y x , B , ℵ ' s and ℜ ' s are
Energy Equation:
τ ∗y x = 

 3Gr 
∂ψ
∂ψ
( x, 0 ) = ( x, 0 ) = 0, θ ( x, 0 ) = θ w ( x )
∂y
∂x
∂ψ
( x, ∞ ) = θ ( x, ∞ ) = 0
∂y
u
 Re  v
, v∗ = 
,

U∞
 3Gr  U ∞
τyx
θw
θ
, θ∗ =
, θ∗ =
,
ρU ∞2
(θ w − θ ∞ ) w (θ w − θ ∞ )
Pr =
U0 L
L
, Gr = 2 g β ' (θ w − θ ∞ )
α ' Re
U∞
where L is the reference length of plate, U 0 is the
reference velocity, U ∞ is the far velocity (near boundary
layer), θ w and θ ∞ are the absolute temperatures of fluid
near plate wall and near boundary layer respectively.
198 ▪ VOL. 44, No 2, 2016
real-valued and at least differential in the real argument
a.
To transform the differential equation, transfor–
mations of the derivatives of ψ can be obtained from G
via chain-rule operations.
a. Invariance analysis
Equations (8) to (10) are said to be invariantly
transformed, for some functions ξi ( a ) whenever,


∂ 2ψ 
∂ 2ψ 
Ω  τ y x , 2  = ξ1 ( a ) Ω τ y x , 2 
∂y 
∂y 


2
2
∂ψ ∂ ψ ∂ψ ∂ ψ ∂
∂ψ
−
− (τ y x ) − θ + bB 2 ( x )
∂y ∂y ∂x ∂x ∂y 2 ∂y
∂y
 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ

 ∂y ∂y∂x − ∂x ∂y 2


= ξ2 ( a ) 
 ∂
∂ψ 
2
 − ∂y (τ y x ) − θ + bB ( x ) ∂y 


∂ψ ∂θ ∂ψ ∂θ
1 ∂ 2θ
−
−
∂y ∂x ∂x ∂y 3Pr ∂y 2
 ∂ψ ∂θ ∂ψ ∂θ
1 ∂ 2θ 
= ξ3 ( a ) 
−
−
2 
 ∂y ∂x ∂x ∂y 3Pr ∂y 
(13)
(14)
(15)
FME Transactions
Substituting the values from (12), using chainrule
operation in above equation, yields
 τ

ℵψ ∂ 2ψ 
∂ 2ψ 
τ
Ω ℵ y xτ y x + ℜ y x ,
=
Ω
ξ
a
τ
,

(
)

 (16)
1
yx

∂y 2 
(ℵy )2 ∂y 2 


and can be obtained by the following first-order linear
partial differential equation: (see Morgan [25], Moran
and Gaggioli [26])
6
∂g
∑ (α Q + β ) ∂Q
i
i
i
i =1
(ℵψ )2  ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ  ℵτ ∂
−
− y
(τ y x )
2 
2 
ℵx (ℵy )  ∂y ∂y∂x ∂x ∂y  ℵ ∂y
= 0, Qi = x, y,ψ , θ ,τ y x , B (23)
i
yx
− (ℵ θ + ℜ
θ
B 2
) + b (ℵ B + R )
θ
B
ℵψ ∂ψ
ℵy ∂y
 ∂ψ ∂ 2ψ ∂ψ ∂ 2ψ
−

∂y ∂y∂x ∂x ∂y 2
= ξ2 ( a ) 
 ∂
2 ∂ψ
 − (τ y x ) − θ + bB
∂y
 ∂y
where,
(17)






ℵψ ℵθ  ∂ψ ∂θ ∂ψ ∂θ 
ℵθ  1 ∂ 2θ 
−
−



x y 
ℵ ℵ  ∂y ∂x ∂x ∂y  (ℵy )2  3Pr ∂y 2 
 ∂ψ ∂θ ∂ψ ∂θ
1 ∂ 2θ 
= ξ3 ( a ) 
−
−
2 
 ∂y ∂x ∂x ∂y 3Pr ∂y 
(18)
= ℜ y = ℜψ = ℜ B = 0,
a = a0
∂ℜi
∂a
The corresponding characteristic equation of (25) is
dx
3dy 3dψ
3dθ dτ yx
3dB
(26)
=
=
=−
=
=−
x
y
2
0
B
+
β
ψ
θ
(
)
where β = β1 / 3α1
Applying the variable separable method, the abso–
lute invariants of independent and dependent variables
owing the equation (25) are given by


= F (η ) 
F1 (η ) = ψ ( x + β )

1/3

F2 (η ) = θ ( x + β ) = G (η ) 

F3 (η ) = τ yx = H (η )

1/3

F4 (η ) = B ( x + β ) = B0

−1/3
−2/3
2
ℵx = (ℵy ) , ℵψ = (ℵy ) , ℵθ =
1
1
τ
, ℵ y x = 1, ℵB = y . (20)
ℵy
ℵ
Finally, we get the one-parameter group G, which
transforms invariantly the system of equations (8)-(10)
along with the auxiliary conditions (11).
 x = y 3 x + ℜ x , y = ℵy y
 (ℵ )
(21)
G :
1
1
2
ψ = (ℵy ) ψ, θ = y θ, B = y B, τ y x = τ y x
ℵ
ℵ

constant say B0 .
The group transformation of absolute invariants is
Now we want to develop a complete set of absolute
invariants so that the original problem (8)–(10) will be
transformed into similarity equations under the derived
deductive group (21).
If η = η ( x, y ) is the absolute invariant of the
independent variables, then variables of the four
absolute invariants for dependent variables ψ, θ, τyx, B
are given by
g j ( x, y ,ψ , θ ,τ y x , B ) = F j (η ) ,
j = 1, 2, 3, 4.
(22)
θ = (x + β )
−1/3
2/3
F (η )
−1/3
G (η )
ψ = (x + β )
b. The complete set of absolute invariants
(27)
As B ( x ) is a functions of x only, F4 (η ) must be
η ( x, y ) = y ( x + β )
FME Transactions
( i = 1,..6 ) (24)
a = a0
∂g  α1 y  ∂g  2α1ψ  ∂g  α1θ  ∂g
+
+
−
∂x  3  ∂y  3  ∂ψ  3  ∂θ
(25)
∂g  α 6 B  ∂B
+ ( 0)
−
=
0.
∂τ y x  3  ∂θ
η ( x, y ) = y ( x + β )
These yield
3
and β i =
(α1 x + β1 )


ψ
ℵ
τyx

ℵ =
= 1 = ξ1 ( a ) ,

(ℵy )2


2
τyx
B 2 ℵψ
ψ
 (19)
(ℵ )
(ℵ )
ℵ
θ
=
=
ℵ
=
=
a
,
ξ

2( )
y
y
2
x
ℵ
ℵ
ℵ (ℵy )

ψ θ
θ

ℵℵ
ℵ

=
= ξ3 ( a ) ,
x y
2
ℵℵ

(ℵy )
τyx
∂ℵi
∂a
and ‘a0’ denotes the value of ‘a’ which yields the
identity element of the group G.
τ
Since ℜθ = ℜ y x = ℜψ = ℜ y = ℜ B = 0 implies that
β1 = β2 = β3 = β4 = β5 = β6 = 0 and from (24) we get
3
α1 = 3α 2 = α 3 = − α 4 = −α 6 , α 5 = 0.
2
Equation (23) yields:
The invariance of equations (16)-(18) together with
boundary conditions (10), implies that
ℜθ = ℜ
αi =
τ yx = H (η )
B = B0 ( x + β )
−1/3










(28)
c. Reduction to ordinary differential equations
Substituting the values of derivatives from (28) in
equations (8)-(10), yields the following system of nonlinear ordinary differential equations.
VOL. 44, No 2, 2016 ▪ 199
( F ′)2 − 2 FF ′ − 3H ′ − G + 3MF ′ = 0.
2 FG ′ + F ′G +



1
G ′ = 0.
Pr
(29)
where Mn = σ B02 ρU ∞ is the magnetic parameter and
Further, the expression of local skin-friction
coefficient C f is:
1
α
=
ReC f ≡ τ yx
sinh −1
y =0
2
γ
{
}
γ f ′′ ( 0 ) . (35)
H (η ) is similarity variable related to non-dimensional
5.
RESULTS AND DISCUSSIONS
strain-stress relation
•
The numerical solutions in a graphical form of nonlinear system (34) subject to the boundary
conditions (31) are obtained using bvp4c solver in
Matlab (Figs. 2-10). This is second order accurate
and allows uniform and non-uniform grid size.
Figures 2-4 are graphical representation of the
profiles similarity variables F', G and F''(0) are
related to velocity along X-axis, local shear stress
and temperature respectively under the influence of
magnetic field M. These figures mean that increase
in M causes the boundary layers to thicken.
At this point it is worth noting that a similar kind of
effects have been observed in the work of Na and
Hansen [12] and T. Hayat et al. [30] for Power-law
and Powel-Eyring Non-Newtonian fluids respectively.
This warrants that the present work is consistent
with earlier work, indeed we have analyze the most
general case and one can studied any NonNewtonian fluid model using present analysis.
Ω ( H , F ′′ ) = 0
(30)
Together with boundary conditions, subject to
1
θ w ( x ) = ( x + β )− 3 are
•
F ( 0 ) = F ′ ( 0 ) = 0, G ( 0 ) = 1,
F ′(∞) = G (∞) = 0
4.
(31)
PRANDTL EYRING FLUID MODEL
•
Non-Newtonian fluid models based on functional
relationship between shear-stress and rate of the strain,
shown by equation (3), are defined by various empirical
explicit or implicit functional relations See [20, 27].
Among these models most research work has been so far
carried out on power-law fluid model, this is because of
its mathematical simplicity. On the other hand fluid
models other than the Power-law model presented in
Table 1 are mathematically more complex and the nature
of partial differential equations governing these flows is
too non-liner boundary value type and hence their
analytical or numerical solution is a bit difficult. For the
present study the Prendtl Eyring model, although mathe–
matically more complex, is chosen mainly due to two
reasons. Firstly, it can be deduced from kinetic theory of
liquids rather than the empirical relation as in the powerlaw model. Secondly, it correctly reduces to Newtonian
behavior for both low and high shear rate. This reason is
somewhat opposite to the pseudo plastic system, whereas
the power-law model has infinite effective viscosity for
the low shear rate, thus limiting its range of applicability.
Mathematically, the Prandtl-Eyring model can be
written as (Bird et al [20], Skelland [27])
 1 ∂u 

 C ∂y 
τ yx = A sinh −1 
•
Figure 2. Magnetic fields effect on velocity profile
(32)
where A and C are flow consistency indices.
Introducing the dimensionless quantities,
H ′ (η ) =
where α =
α F ′′′
2 1/2
{1 + γ ( F ′′) }
,
(33)
Figure 3. Magnetic field effects on local shear-stress profile
3
ρU ∞
A
, γ =
are dimensionless flow
µ∞ C
µ LC 2
parameter.
Substituting it into the equation (29), we get
(
1
F ′2 − 2 FF ′′ − G − 3MF ′
3α
1
2 FG ′ + F ′G + G ′ = 0.
Pr
F ′′′ =
200 ▪ VOL. 44, No 2, 2016
)

1 + γ F ′′2 .
 (34)



Figure 4. Magnetic field effects on temperature profile
FME Transactions
5(a)
5(b)
Figure 5. Influence of flow parameter on non-dimensional
velocity
6(a)
7(a)
7(b)
Figure 7. Influence of flow parameter on non-dimensional
temperature
Figure 8. Effect of Prandtl number on dimensionless
velocity of Prandtl Eyring fluid
6(b)
Figure 6. Influence of flow parameter on non-dimensional
local shear-stress
FME Transactions
Figure 9. Effect of Prandtl number on skin-friction
coefficient of Prandtl Eyring fluid
VOL. 44, No 2, 2016 ▪ 201
REFERENCES
Figure 10. Temperature variation under the effect of Prandtl
number on Prandtl Eyring fluid
•
•
•
•
6.
Figs. 5-7 display the effect of flow parameters α
and γ on similarity functions variables related to
velocity along X direction, local skin-friction and
temperature. These figures depict boundary layer
thickness decrease as the flow parameter α
increases by controlling the flow parameter γ and
M. A similar trend can be observed for another flow
parameter. This behavior of flow is consistent with
Hayat et al. [30].
Also, the rheological parameter B and µ∞ have an
inverse relation with dimensionless parameter.
Decrease in one of the rheological parameters means
an increase in one of the flow parameter. Figs. 8-10
represent the effect of magnetic parameters, hence
the decrease of boundary layer thickness.
Figs. 8-10 display the influence of Prandtl number
on fluid flow. These figures depict an increase in
Prandtl number causing the boundary and thermal
boundary layer thichness. The present results are
the same as those mentioned in S. Panigrahi et al.
[31]. This comparision gurranteed the validity of
present analysus.
It is worth noting that all solutions have derived for
non-dimensional quantities and hence these results
are applicable for all types of considered nonNewtonian fluids.
CONCLUSION
The deductive group symmetry method is applied to
search similarity transformations to transform the
partial differential system to ordinary differential
system for the class of Non-Newtonian fluids.
Numerical solutions are presented in a graphical form
for Prandtl-Eyring fluids using Matlab. Effects of
rheological parameters on the boundary layers are
discussed in detail. It is found that change in all the
dimensionless parameters and rheological parameters
causes the boundary layers thickness. The analysis is
made for generalized Non-Newtonian fluid and work
of Na and Hansen [12] and T. Hayat et al. [30] are
particular cases of he present work.
ACKNOWLEDGMENT
The authors express their gratitude to the unknown
referees for their critical reviews and fruitful comments.
202 ▪ VOL. 44, No 2, 2016
[1] Oosthuizen, P.H. An introduction to convective
Heat Transfer Analysis, McGraw-Hill. 1999.
[2] Soundalgekar, V. M., Free convection effects on
the oscillatory flow past an infinite, vertical porous
plate with constant suction II, Proc. R. Soc.
London, Vol. A 333, pp 37-50. 1973,
[3] Soundalgekar, V. M., Free convection effects on the
Stokes problem for an infinite vertical plate, J. Heat
transfer, Trans. ASME, Vol 99, pp 499-501, 1977.
[4] Georgantopoulos, G. et al.: Free convection and
mass transfer effects on the oscillatory flow of a
dissipative fluid past an infinite vertical porous
plate II, Czechoslovac. Jour. of Phy., B 31, pp 876884, 1981.
[5] Revankar, S. T., Natural convection effects on
MHD flow past an impulsively started permeable
vertical plate, Ind. J. Pure and Appl. Maths, Vol
14(4), pp 530-539, 1983.
[6] Soundalgekar, V. M., Free convection effects on the
Stokes problem for an infinite vertical plate, J. Heat
transfer, Trans. ASME, Vol. 99, pp 499-501, 1977.
[7] Kafousias, G., MHD thermal –diffusion effects on
free convective and mass transfer flow over an
infinite vertical moving plate, Astro.Phy.Space Sci,
Vol. 192. pp 11-19, 1992.
[8] Ingham D. B, Pop I. (eds.), Transport Phenomena
in Porous Media. Pergamon, Oxford., vol. I, 1998,
vol. II, 2002.
[9] Nakayama, A., Shenoy, A. V., Combined forced
and free convection heat transfer in Power-Law
fluid-saturated porous media. Appl. Sci. Res. Vol.
50, pp. 83-95, 1993.
[10] Lee, S. Y. and Ames. W. F., Similarity solutions
for non-Newtonian fluids. AIChE JI, Vol. 12,
pp.700-708, 1966.
[11] Acrivos, A., A theoretical analysis of laminar
natural convection heat transfer to Non-Newtonian
fluids, AIChE. J., vol 6, no. 4, 584-590, 1960.
[12] Na, T. Y. and Hansen, A. G., Possible similarity
solution of laminar natural convection flow of nonNewtonian fluids, Int. J. of heat and mass transfer,
Vol 9, pp 261, 1966.
[13] Sarpakaya, T., Flow of non-Newtonian fluids in a
magnetic field, AICh Journal Vol. 7, pp. 324-328,
19961.
[14] Andersson, H. I., Bech, K. H. and Dandapat, B. S.,
Magnetohydrodynamic flow of a power-law fluid
over a stretching sheet, Int. J. Non-Linear Mech.
Vol. 27, pp. 929-936, 1992.
[15] Martinson, L. K., Pavlov, K. B., Unsteady shear
flows of a conducting fluid with a rheological
power law, MHD Vol. 7, pp. 182-189, 1971.
[16] Samokhen, V. N., On the boundary-layer equation
of MHD of dilatant fluid in a transverse magnetic
field, MHD Vol. 3, pp. 71-77, 1987.
[17] Cortell, R., A note on magneto hydrodynamic flow
of a power law fluid over a stretching sheet, Appl.
Math. Comput. Vol. 168, pp. 557-566, 2005.
FME Transactions
[18] Saponkov, Y., Similar solutions of steady boundary
layer equations in magnetohydrodynamic power
law conducting fluids, Mech. Fluid Gas. Vol. 6 pp.
77-82, 1967.
[19] Liao Shi-Jun, On the analytic solution of
magnetohydrodynamic flows of non-Newtonian
fluids over a stretching sheet, J. Fluid Mech. Vol.
488, pp. 189-212, 2005.
[20] Bird, R. B., Stewart, W. E. and Lightfoot, E. M.
Transport phenomena, John Wiley, New York, 1960.
[21] Abd-el-Malek, et al., Solution of the Rayleigh
problem for a power law non-Newtonian
conducting fluid via group method. Int. J. Eng. Sci.
Vol. 40, pp. 1599-1609, 2002.
[22] Adnan, K. A., Hasmani, A. H. and Timol, M. G., A
new family of similarity solutions of three
dimensional MHD boundary layer flows of nonNewtonian fluids using new systematic grouptheoretic approach, Applied Mathematical Sciences
Vol. 5, No.(27, pp. 1325-1336, 2011.
[23] Darji, R. M. and Timol, M. G., Deductive Group
Theoretic Analysis for MHD Flow of a Sisko Fluid
in a Porous Medium, Int. J. of Appl. Math and
Mech. Vol. 7, No. 19, pp. 49-58, 2011.
[24] Darji, R. M. and Timol, M. G., Invariance analysis
and similarity solution of heat transfer for MHD
viscous power-law fluid over a non-linear porous
surface via group theory, Int. J. Adv. Appl. Math.
and Mech. Vol. 1, No. 2, pp. 116-132, 2013.
[25] Morgan A. J. A., The reduction by one of the
number of independent variables in some systems
of nonlinear partial differential equations. Quart. J.
Math. Oxford Vol. 3, pp. 250–259, 1952.
[26] Moran, M. J. and Gaggioli, R. A., Reduction of the
number of variables in system of partial differential
equations with auxiliary conditions. SIAM J. Appl.
Math. Vol. 16, pp. 202-215, 1968.
[27] Skelland, A. H. P., Non-Newtonian flow and heat
transfer, John Wiley, New York, 1967.
[28] Gebhart B. and Pera, L., The nature of vertical
natural convection flows resulting from the
combined buoyancy effects of thermal and mass
diffusion, Int. J. Heat Mass Transfer, Vol. 14, pp.
2025–2049, Zbl 223.76060, 1971.
[29] Gebhart, B., et al., Bouyancy-Induced Flows and
Transport, Hemisphere Publishing Corporation,
1988. Zbl 699.76001.
[30] Hayat, T., et al., Similarity solutions for boundary
layer equations of a Powel-Eyring fluid,
Mathematical and Computational Applications,
Vol. 18, No. 1, 62-70, 2013.
FME Transactions
[31] Panigrahi, S., Reza, M. and Mishra. A. K., MHD
effect of mixed convection boundary-layer flow of
Powell-Eyring fluid past nonlinear stretching
surface, Applied Mathematics and Mechanics, Vol.
35, No. 12, pp. 1525-1540, 2014.
NOMENCLATURE
A, C
B
F , G, H
g
Gr
Pr
Re
u, v
U
x, y
Flow consistency indices
Magnetic field strength
Transformed dependent variables
Gravitational force
Grashof number
Prandtl number
Reynolds number
Components of velocity in x, y direction
Main stream velocity
Rectangular coordinates
Greek symbols
α'
β'
ρ
v
µ
ψ
τ
Ω
η
θ
θw
Thermal diffusivity
Volumetric thermal expansion coefficient
Density
Kinematics viscosity
Viscosity
Stream function
Shearing stress
Functional relation of τ and velocity
gradient
Transformed independent variables
Temperature distribution
Temperature distribution near plate wall
ПОСТУПАК УТВРЂИВАЊА СЛИЧНОСТИ
КОД СЛОБОДНОГ КОНВЕКТИВНОГ МХД
СТРУЈАЊА У ГРАНИЧНОМ СЛОЈУ КЛАСЕ
НЕЊУТНОВСКОГ ФЛУИДА
Р.М. Дарји, М.Г. Тимол
У раду се користи поступак симетрије дедуктивне
групе за извођење трансформација сличности код
слободног конвективног струјања у граничном слоју
класе нењутновских флуида који прелазе преко
дводимензионалне површине и теку под утицајем
попречног магнетног поља. Нумеричка решења су
добијена у графичком облику за одређени модел
нењутновског флуида, тј. Прандтл-Ерингов флуид.
Разматрају се значајне физичке величине као што је
дистрибуција брзине, коефицијент спољашњег
трења и температурне варијације.
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